Math Insight

Thread: Calculus Refresher

A brief review of the concepts of single variable calculus

1. Preliminaries
   • Introduction to Calculus Refresher
   • Polynomial inequalities
   • Domain of functions
   • Lines (and other items in Analytic Geometry)
   • Elementary limits
   • Limits with cancellation
   • Limits at infinity
   • Limits of exponential functions at infinity
2. Differentiation
   • The idea of the derivative of a function
   • Derivatives of polynomials
   • Derivatives of more general power functions
   • The quotient rule
   • The product rule
   • The chain rule
   • Tangent and normal lines
   • Critical points, monotone increase and decrease
   • Minimization and maximization refresher
   • Local minima and maxima (First Derivative Test)
   • An algebra trick for finding critical points
   • Linear approximations: approximation by differentials
   • Implicit differentiation
   • Related rates
   • Intermediate Value Theorem, location of roots
   • Newton's Method
   • Derivatives of transcendental functions
   • L'Hospital's rule
   • Exponential growth and decay: a differential equation
   • The second and higher derivatives
   • Inflection points, concavity upward and downward
   • Another differential equation: projectile motion
   • Graphing rational functions, asymptotes
3. Integration
   • Basic integration formulas
   • The simplest integration substitutions
   • Integration substitutions
   • Area and definite integrals
   • Length of curves
   • Numerical integration
   • Averages and weighted averages
   • Centers of mass (centroids)
   • Volumes by cross sections
   • Volume of surfaces of revolution
   • Integration by parts
   • Partial fractions
   • Trigonometric integrals
   • Trigonometric substitution
4. Taylor polynomials
   • Historical and theoretical comments: Mean Value Theorem
   • Taylor polynomials: formulas
   • Classic examples of Taylor polynomials
   • Computational tricks regarding Taylor polynomials
   • Prototypes: More serious questions about Taylor polynomials
   • Determining tolerance/error in Taylor polynomials.
   • How large an interval with given tolerance for a Taylor polynomial?
   • Achieving desired tolerance of a Taylor polynomial on desired interval
   • Integrating Taylor polynomials: first example
   • Integrating the error term of a Taylor polynomial: example

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